[FOM] Boolean Rings : Commutative Rings :: Propositional Logic : A 'free' Logic?
a_mani_sc_gs at yahoo.co.in
Thu May 21 23:08:41 EDT 2009
--- On Fri, 22/5/09, Vaughan Pratt <pratt at cs.stanford.edu> wrote:
> The earliest proposal I'm aware of to drop x^2 = x,
> Contraction, without
> also dropping commutativity is Girard's linear logic,
> presented as
> invited talks at LICS'86 (the first LICS, held in Cambridge
> MA) and a
Linear logics and related logics (up to Full Lambek) that are related to residuated lattices do not seem close to the answer as they assume too much. A CL algebra does not fit. But the motivation for a quantitative conjunction is there.
A simple examination of the lattice of varieties for the problem will not yield a definite answer. Many abstract logics may fit in. But quantum logics would be closest and there is no clear minimal logic for the question. The logics with semiring+ algebraic semantics may be further away, but would be more useful.
Vakarelov's papers "Non-Classical Negation in the Works of Helena Rasiowa and Their Impact on the Theory of Negation. Studia Logica, vol 84, No 1, 2006, 105-127.
Consistency, Completeness and Negation. In: Paraconsistent Logic. Essays on the Inconsistent. Gr. Priest, R. Routley and J. Norman Eds. Analitica, Philosophia Verlag, Munhen, 1989."
make the relation clear from a proof theoretical point of view.
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